Radial power integrability criterion
= Radial power integrability criterion
If a radial function on $\mathbb R^n$ behaves like $r^a$ near zero and like $r^{-b}$ at infinity, then its $q$th power is locally integrable at zero exactly when $aq>-n$ and integrable at infinity exactly when $bq>n$. This follows from the radial measure factor $r^{n-1}\,dr$.